$$\require{cancel}$$

# 3.11: Exercises

1. Monochromatic light with a wavelength of $$6000 \unicode{x212b}$$ passes through a fast shutter that opens for $$10^{-9}$$ sec. What is the subsequent spread in wavelengths of the no longer monochromatic light?
2. Calculate $$\langle x\rangle$$, $$\langle x^{\,2}\rangle$$, and $$\sigma_x$$, as well as $$\langle p\rangle$$, $$\langle p^{\,2}\rangle$$, and $$\sigma_p$$, for the normalized wavefunction $\psi(x) = \sqrt{\frac{2\,a^{\,3}}{\pi}}\,\frac{1}{x^{\,2}+a^{\,2}}.$ Use these to find $$\sigma_x\,\sigma_p$$. Note that $$\int_{-\infty}^{\infty} dx/(x^{\,2}+a^{\,2}) = \pi/a$$.
3. Classically, if a particle is not observed then the probability of finding it in a one-dimensional box of length $$L$$, which extends from $$x=0$$ to $$x=L$$, is a constant $$1/L$$ per unit length. Show that the classical expectation value of $$x$$ is $$L/2$$, the expectation value of $$x^{\,2}$$ is $$L^2/3$$, and the standard deviation of $$x$$ is $$L/\sqrt{12}$$.
4. Demonstrate that if a particle in a one-dimensional stationary state is bound then the expectation value of its momentum must be zero.
5. Suppose that $$V(x)$$ is complex. Obtain an expression for $$\partial P(x,t)/\partial t$$ and $$d/dt \int P(x,t)\,dx$$ from Schrödinger’s equation. What does this tell us about a complex $$V(x)$$?
6. $$\psi_1(x)$$ and $$\psi_2(x)$$ are normalized eigenfunctions corresponding to the same eigenvalue. If $\int_{-\infty}^\infty \psi_1^\ast\,\psi_2\,dx = c,$ where $$c$$ is real, find normalized linear combinations of $$\psi_1$$ and $$\psi_2$$ that are orthogonal to (a) $$\psi_1$$, (b) $$\psi_1+\psi_2$$.
7. Demonstrate that $$p=-{\rm i}\,\hbar\,\partial/\partial x$$ is an Hermitian operator. Find the Hermitian conjugate of $$a = x + {\rm i}\,p$$.
8. An operator $$A$$, corresponding to a physical quantity $$\alpha$$, has two normalized eigenfunctions $$\psi_1(x)$$ and $$\psi_2(x)$$, with eigenvalues $$a_1$$ and $$a_2$$. An operator $$B$$, corresponding to another physical quantity $$\beta$$, has normalized eigenfunctions $$\phi_1(x)$$ and $$\phi_2(x)$$, with eigenvalues $$b_1$$ and $$b_2$$. The eigenfunctions are related via \begin{aligned} \psi_1 &= (2\,\phi_1+3\,\phi_2) \left/ \sqrt{13},\right.\nonumber\\[0.5ex] \psi_2 &= (3\,\phi_1-2\,\phi_2) \left/ \sqrt{13}.\right.\nonumber\end{aligned} $$\alpha$$ is measured and the value $$a_1$$ is obtained. If $$\beta$$ is then measured and then $$\alpha$$ again, show that the probability of obtaining $$a_1$$ a second time is $$97/169$$.
9. Demonstrate that an operator that commutes with the Hamiltonian, and contains no explicit time dependence, has an expectation value that is constant in time.
10. For a certain system, the operator corresponding to the physical quantity $$A$$ does not commute with the Hamiltonian. It has eigenvalues $$a_1$$ and $$a_2$$, corresponding to properly normalized eigenfunctions \begin{aligned} \phi_1 &= (u_1+u_2)\left/\sqrt{2},\right.\nonumber\\[0.5ex] \phi_2 &= (u_1-u_2)\left/\sqrt{2},\right.\nonumber\end{aligned} where $$u_1$$ and $$u_2$$ are properly normalized eigenfunctions of the Hamiltonian with eigenvalues $$E_1$$ and $$E_2$$. If the system is in the state $$\psi=\phi_1$$ at time $$t=0$$, show that the expectation value of $$A$$ at time $$t$$ is $\langle A\rangle = \left(\frac{a_1+a_2}{2}\right) + \left(\frac{a_1-a_2}{2}\right)\cos\left(\frac{[E_1-E_2]\,t}{\hbar}\right).$

## Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
